The answer
Specifically, the arguments to this jacobian are the state of the robot.
The reason
It is the jacobian of the measurement function with respect to the landmark state.
If you knew the state of the robot and landmark, what function would you use to predict what the measurement would be? If you have a range sensor, it would be the distance between the landmark and robot positions. Take the jacobian of this function, with respect to each of the state variables (robot x, robot y, robot theta, landmark x, landmark y)
In their first example, we're measuring the difference in x and y.
So the predicted position of the landmark equals the position of the robot plus the $x$ and $y$ difference (rotated a little by the robot rotations).
So the jacobian is $$\begin{bmatrix}
\frac{d h_1}{dx_r}, \frac{d h1}{dy_r}, \frac{d h1}{d\theta_r}\\
\frac{d h_2}{dx_r}, \frac{d h2}{dy_r}, \frac{d h2}{d\theta_r}\\
\end{bmatrix}$$
Here, $h1$ is the function which gives the $x$ coordinate of the landmark, and $h2$ is the function which gives the $y$ coordinate of the landmark.